bài tập tổng hợp đại số tuyến tính

BÀI TẬP ĐẠI SỐ TUYẾN TÍNH Tính A + B, AB, BA biết ⎡1 ⎤ ⎡ −4 ⎤ , B=⎢ a A = ⎢ ⎥ ⎥ ⎣3 ⎦ ⎣2 ⎦ ⎡ −1 ⎤ ⎡ −2 ⎤ ⎢ ⎥ b A = ⎢ 1 ⎥ , B = ⎢⎢ ⎥⎥ ⎢⎣ −1 ⎥⎦ ⎢⎣ −3 ⎥⎦ Đáp số ⎡ −1 ⎤ ⎡ −1 −3 ⎢ ⎥ b A + B = ⎢ ⎥ , AB = ⎢⎢ −2 ⎢⎣ −4 ⎥⎦ ⎢⎣ −2 −7 Tính tích ma trận ⎡5 ⎤ ⎡ −2 ⎤ ⎛ ⎡ ⎜ a ⎢⎢5 ⎥⎥ ⎢⎢ −3 −4 −5⎥⎥ ⎜ ⎢⎢ 10 ⎢⎣6 ⎥⎦ ⎢⎣ ⎥⎦ ⎜⎝ ⎢⎣ −5 ⎡3 b ⎢⎢ −1 ⎢⎣ ⎡1 ⎢1 c ⎢ ⎢1 −2 ⎢ ⎣1 ⎡2 ⎢4 d ⎢ ⎢ −2 ⎢ ⎣1 −3⎤ 5⎤ ⎡ ⎥ ⎢ 16 ⎥ , BA = ⎢ 23 −1 14 ⎥⎥ ⎢⎣ −17 −12 11⎥⎦ 15 ⎥⎦ ⎤⎞ ⎟ ⎥⎥ ⎟ −7 ⎥⎦ ⎟⎠ ⎛ ⎡ 11 13 ⎤ ⎞ ⎜⎢ ⎥⎟ ⎜ ⎢ −22 −27 −17 ⎥ ⎟ ⎜ ⎢ 29 32 26 ⎥⎦ ⎟⎠ ⎝⎣ ⎛ ⎡ −1 ⎤ ⎞ ⎜⎢ ⎥⎟ ⎜⎢ 6 ⎥⎟ ⎜ ⎢12 −3 20 ⎥ ⎟ ⎜⎜ ⎢ ⎥ ⎟⎟ ⎦⎠ ⎝⎣ 9⎤ ⎡ ⎤ ⎥⎥ ⎢⎢ ⎥⎥ ⎥⎦ ⎢⎣ −4 −5 −3⎥⎦ −2 ⎤ ⎡1 −2 ⎤ −1⎥⎥ ⎢ ⎥⎥ ⎢ 5⎥ ⎥ ⎢ ⎥⎦ −2 ⎦ ⎣ 3⎤ ⎛ ⎡1 ⎤ ⎞ ⎡ ⎤ ⎜ ⎟ ⎥⎥ ⎢ ⎥ ⎜ ⎢7 ⎥ ⎟ ⎢ ⎥ −3⎥ ⎢ ⎥ ⎜ ⎢ ⎥ ⎟ ⎥ ⎢ −1⎥ ⎜ ⎢ ⎥ ⎟ ⎦ ⎣ ⎦ ⎜⎝ ⎣9 ⎦ ⎟⎠ ⎡1 ⎤ ⎢ ⎥ ⎡1 −3 −1⎤ ⎢1 ⎥ ⎛ ⎡0 ⎤ ⎞ e ⎢ ⎜⎢ ⎥⎟ ⎥ ⎣1 −5 ⎦ ⎢1 ⎥ ⎝ ⎣0 ⎦ ⎠ ⎢ ⎥ ⎣1 −2 ⎦ ⎛ ⎡3 ⎤ ⎞ ⎡1 ⎤ ⎜ ⎟ ⎢ ⎥ f ⎢ ⎥ [3 1] ⎜ ⎢⎢6 ⎥⎥ ⎟ ⎜ ⎢9 ⎥ ⎟ ⎢⎣ ⎥⎦ ⎦⎠ ⎝⎣ Tính tích AB, BA ⎡ −1 ⎢ −2 = a A ⎢ ⎢3 ⎢ ⎣4 0⎤ ⎥⎥ ⎡ −1 ⎤ ⎛ ⎡10 15 −5⎤ ⎞ , B=⎢ ⎜ BA = ⎢ ⎥ ⎥⎟ −2 ⎥ ⎣ −1 ⎦ ⎝ ⎣11 10 10 ⎦ ⎠ ⎥ 2⎦ ⎡2 ⎤ ⎢1 −4 ⎥ ⎥ , B = [5 −3] ( BA = [11 −1]) b A = ⎢ ⎢3 ⎥ ⎢ ⎥ ⎣ −1⎦ ⎡1 ⎤ ⎢6 ⎥ ⎛ ⎡1 4⎤ ⎥ ⎜ AB = ⎡ 28 27 ⎤ ⎟⎞ , B=⎢ c A = ⎢ ⎢15 14 13⎥ ⎥ ⎢1 −1⎥ ⎝ ⎣ ⎦⎠ ⎣ −2 ⎦ ⎢ ⎥ ⎣3 ⎦ Tính định thức sau 0 a 0 b a Δ = ( abcd ) c 0 d 0 b Δ = c Δ = 1 4 3 2 1 1 3 −1 d Δ = e Δ = 1 ( 0) ( −1) 3 −1 ( −264 ) 2 −1 −1 −1 −2 −2 ( −12 ) −1 −2 f Δ = 4 −1 −1 ( 24 ) −1 −2 g Δ = h Δ = i Δ = j Δ = k Δ = l Δ = a b ( abcd ) c 0 d 1 a 2 b c d a 1 b 1 1 1 c d ( 4a − c − d ) ( 2a + b − c + d ) −1 −1 −2 a b −2 −1 c d ( −5a − 5b − 5c − 5d ) 3 −5 −4 −4 −2 −5 −4 −1 −4 −8 −1 ( −2858) ( −264 ) 0 m Δ = 2 (1) 3 n Δ = 0 0 0 ( 4) o Δ = p Δ = 0 0 0 0 ( −21) −1 −2 −3 ( 0) 2 1 1 1 q Δ = 0 ( ) 0 0 Tìm dạng bậc thang hạng ma trận ⎡ −1 0 ⎤ ⎢ 1 2⎥ ⎢ ⎥ a A = ⎢ 1 1 ⎥ ( r ( A ) = 3) ⎢ ⎥ ⎢ 1⎥ ⎢⎣ ⎥⎦ ⎡ −3 ⎤ ⎢ −2 ⎥ ⎢ ⎥ b A = ⎢ −2 11⎥ ( r ( A ) = ) ⎢ ⎥ ⎢ −15 −7 ⎥ ⎢⎣ −1 ⎥⎦ ⎡1 ⎤ c A = ⎢⎢3 −1 −2 ⎥⎥ ( r ( A ) = ) ⎢⎣5 10 ⎥⎦ ⎡1 ⎢2 d A = ⎢ ⎢0 ⎢ ⎣1 1 11 14 ⎡1 ⎢2 e A = ⎢ ⎢ −2 −5 ⎢ ⎣1 4 2⎤ ⎥⎥ ( r ( A) = ) 1⎥ ⎥ 3⎦ 5⎤ 12 ⎥⎥ ( r ( A ) = 3) 5⎥ ⎥ 20 ⎦ ⎡0 f A = ⎢⎢ ⎢⎣ ⎡1 g A = ⎢⎢ ⎢⎣ 3⎤ −1⎥⎥ ( ) ⎥⎦ −2 ⎤ −4 ⎥⎥ ( ) ⎥⎦ ⎡1 2⎤ h A = ⎢⎢ ⎥⎥ (1) ⎢⎣ −1 −2 ⎥⎦ ⎡1 i A = ⎢⎢ −1 ⎢⎣ ⎡ −1 j A = ⎢⎢ ⎢⎣ −2 ⎤ ⎥⎥ ( 3) −1 ⎥⎦ −2 ⎤ −1 −3⎥⎥ ( ) −8⎥⎦ ⎡ −2 ⎤ ⎢ −3 −1⎥ ⎥ ( 3) k A = ⎢ ⎢5 1⎥ ⎢ ⎥ ⎣1 0⎦ ⎡4 2⎤ ⎢ −1 ⎥ ⎥ ( 4) l A = ⎢ ⎢ −1 −3⎥ ⎢ ⎥ ⎣ −3 −1 ⎦ ⎡ −1 −3 −2 −3⎤ ⎢ 4 −1⎥ ⎥ ( 3) m A = ⎢ ⎢ −6 −1 −1 ⎥ ⎢ ⎥ ⎣ 12 −4 ⎦ Tìm ma trận nghịch đảo (nếu có) ⎡ −2 ⎤ ⎛ ⎡ −5 ⎤ ⎞ ⎜1 ⎟ a A = ⎢⎢1 −3 ⎥⎥ ⎜ ⎢⎢15 −8 ⎥⎥ ⎟ 10 ⎢⎣6 −3⎥⎦ ⎜⎝ ⎢⎣ 25 −14 ⎥⎦ ⎟⎠ ⎡ −2 ⎢ −5 ⎢ b A = ⎢ ⎢ ⎢2 ⎢⎣ −2 1 −7 ⎤ ⎥⎥ 9⎥ ⎥ 2⎥ ⎥⎦ ⎛⎡ ⎜⎢ ⎡ ⎤ ⎜⎢ c A = ⎢⎢ −1 −2 ⎥⎥ ⎜⎜ ⎢ − ⎢ ⎢⎣ −1 ⎥⎦ ⎜ ⎢ ⎜⎢ ⎜⎢ ⎝⎣ ⎡2 d A = ⎢⎢ ⎢⎣ ⎡1 −1 e A = ⎢⎢5 ⎢⎣1 4 − ⎤⎞ ⎥ ⎟⎟ ⎥ ⎥⎟ ⎥⎟ ⎟ ⎥⎥ ⎟ − ⎟ ⎥⎦ ⎠ ⎛⎡ ⎤⎞ ⎜⎢ − ⎥⎟ ⎤ ⎜⎢ ⎥⎟ 1⎥ ⎟ ⎥ ⎜ ⎢ 8⎥⎜ − − ⎟ ⎢ 3 3⎥ 12 ⎥⎦ ⎜ ⎢ ⎥⎟ ⎜⎢ − 1 ⎥⎟ ⎜⎢ 4 ⎦⎥ ⎟⎠ ⎝⎣ ⎤ ⎛ ⎡ −9 11 −5⎤ ⎞ ⎜ ⎟ ⎥⎥ ⎜ ⎢⎢ −4 13 ⎥⎥ ⎟ 41 −1⎥⎦ ⎜⎝ ⎢⎣19 −5 ⎥⎦ ⎟⎠ ⎡1 −3⎤ f A = ⎢⎢3 −1 ⎥⎥ ⎢⎣5 −1⎥⎦ ⎡ −3 ⎤ ⎛ ⎡13 15 −12 ⎤ ⎞ ⎜ ⎢ ⎟ ⎢ ⎥ g A = ⎢ −1 ⎥ ⎜ − ⎢17 10 −8 ⎥⎥ ⎟ 25 ⎢⎣ −2 ⎥⎦ ⎜⎝ ⎢⎣ −1 −5 −1 ⎥⎦ ⎟⎠ ⎡ 1 ⎤ ⎛ ⎡ −3 ⎤ ⎞ ⎜1 ⎟ h A = ⎢⎢ 0 ⎥⎥ ⎜ ⎢⎢ −2 −2 ⎥⎥ ⎟ ⎢⎣ −1 ⎥⎦ ⎜⎝ ⎢⎣ ⎥⎦ ⎟⎠ ⎡ −1 i A = ⎢⎢ ⎢⎣ ⎡3 j A = ⎢⎢ ⎢⎣ 4⎤ ⎛ ⎡ −11 ⎤ ⎞ ⎜ ⎟ ⎥⎥ ⎜ − ⎢0 −5 ⎥ ⎟ ⎥ 3⎢ ⎢⎣0 −1⎥⎦ ⎟⎠ ⎥⎦ ⎜⎝ −1⎤ ⎛ ⎡ −12 ⎤ ⎞ ⎜ ⎟ ⎥⎥ ⎜ ⎢ −1 17 −7 ⎥ ⎟ ⎢ ⎥ ⎥⎦ ⎜⎝ ⎢⎣ −2 ⎥⎦ ⎟⎠ ⎡ ⎢1 ⎢ ⎢ k A = ⎢ ⎢ ⎢0 − ⎢⎣ ⎤ ⎥ ⎥ ⎥ 2⎥ ⎥ ⎥ ⎥⎦ ⎛⎡ ⎜ ⎢1 ⎜⎢ ⎜⎢ ⎜ ⎢0 ⎜⎢ ⎜⎢ ⎜ ⎢0 ⎝⎣ 2 ⎤⎞ ⎥⎟ ⎥⎟ ⎥⎟ − ⎟ ⎥⎟ ⎥ ⎥⎟ ⎥⎦ ⎟⎠ ⎡1 2 ⎤ ⎛ ⎡1 ⎜1 l A = ⎢⎢ −2 ⎥⎥ ⎜ ⎢⎢ ⎢⎣ −2 ⎥⎦ ⎜⎝ ⎢⎣ −2 ⎡ −1⎤ ⎛ ⎡ −1 ⎜1 m A = ⎢⎢ −2 ⎥⎥ ⎜ ⎢⎢ −3 ⎢⎣ ⎥⎦ ⎜⎝ ⎢⎣ −1 ⎡ −1 ⎤ ⎛ ⎡ −2 ⎜1 n A = ⎢⎢ ⎥⎥ ⎜ ⎢⎢ ⎢⎣ ⎥⎦ ⎜⎝ ⎢⎣ −7 ⎡1 o A = ⎢⎢ ⎢⎣ −1 ⎡1 p A = ⎢⎢ −1 ⎢⎣ 1⎤ ⎛ ⎡ ⎜1 ⎥⎥ ⎜ ⎢ 9⎢ 2 ⎥⎦ ⎜⎝ ⎢⎣ −1 −3 −1⎤ ⎛ ⎡17 ⎜ ⎥⎥ ⎜ ⎢ ⎢ −2 ⎥⎦ ⎜⎝ ⎢⎣13 −1 15 12 ⎤⎞ ⎟ −2 ⎥⎥ ⎟ ⎥⎦ ⎟⎠ ⎤⎞ ⎟ −1⎥⎥ ⎟ ⎥⎦ ⎟⎠ ⎤⎞ ⎟ −3⎥⎥ ⎟ ⎥⎦ ⎟⎠ −5⎤ ⎞ ⎟ ⎥⎥ ⎟ ⎥⎦ ⎟⎠ −1⎤ ⎞ ⎟ ⎥⎥ ⎟ −1⎥⎦ ⎟⎠ Tìm a để ma trận au khả nghịch ⎡ −2 ⎤ 9⎞ ⎛ a A = ⎢⎢ a ⎥⎥ ⎜ a ≠ ⎟ 4⎠ ⎝ ⎢⎣ 1 ⎥⎦ ⎡a ⎤ b A = ⎢⎢ a ⎥⎥ a ∉ 0; − 5; ⎢⎣ a ⎥⎦ Tìm ma trận X thỏa phương trình sau ( { ⎡ −1⎤ ⎡1 −1⎤ ⎛ ⎡ }) 0⎤ ⎞ a ⎢ ⎥ X = ⎢ ⎥ ⎜ ⎢ −3 ⎥ ⎟ ⎣3 ⎦ ⎣ ⎦⎝ ⎣ ⎦⎠ ⎡ 1⎤ ⎡ −2 ⎤ ⎥=⎢ ⎥ ⎣ 1⎦ ⎣ −1⎦ b X ⎢ ⎛ ⎡ −5 ⎤ ⎞ ⎜⎢ ⎥⎟ ⎝ ⎣13 −5⎦ ⎠ ⎡ −1⎤ ⎡ ⎤ ⎡1 −1⎤ ⎛ ⎡ ⎥X ⎢ ⎥=⎢ ⎥ ⎜⎢ ⎣ ⎦ ⎣ −1 ⎦ ⎣3 ⎦ ⎝ ⎣11 ⎡1 1⎤ ⎡ −1 ⎤ ⎡2 ⎢ ⎥ ⎢ ⎥ d ⎢0 1⎥ X + ⎢ ⎥ = ⎢⎢ ⎢⎣ 0 1⎥⎦ ⎢⎣ −2 −1⎥⎦ ⎢⎣1 c ⎢ −3⎤ ⎞ ⎟ −2 ⎥⎦ ⎠ ⎤ ⎛ ⎡ −5 16 −8⎤ ⎞ ⎜ ⎟ −3 ⎥⎥ ⎜ ⎢⎢ −7 ⎥⎥ ⎟ −1 ⎥⎦ ⎜⎝ ⎢⎣ −2 ⎥⎦ ⎟⎠ ⎡ −1 ⎤ ⎡ −2 ⎤ ⎡1 0 ⎤ ⎛ ⎡ −21 45 −156 ⎤ ⎞ ⎜1 ⎟ ⎢ ⎥ e X ⎢ −2 ⎥ − ⎢⎢ −1 ⎥⎥ = ⎢⎢0 0⎥⎥ ⎜ ⎢⎢ −21 15 −21 ⎥⎥ ⎟ 15 ⎢⎣ −1 ⎦⎥ ⎣⎢ 0 ⎦⎥ ⎜⎝ ⎣⎢ 51 20 −79 ⎦⎥ ⎟⎠ ⎣⎢ −1 ⎦⎥ Giải hệ phương trình sau ⎧ x1 − x2 + x3 − x4 = ⎪ x + x2 − x3 + x4 = a ⎪⎨ ( 2; −1;0; −2 ) ⎪3x1 + x2 − x3 − x4 = ⎪⎩ x1 − x2 + x3 + x4 = −7 ⎧2 x1 − x2 + x3 + x4 = ⎪ x + x2 − x3 + x4 = ⎛ ⎞ b ⎪⎨ ⎜ ; − ;1; ⎟ ⎝3 3⎠ ⎪5 x1 + x2 + 3x3 = ⎪⎩3x1 − x2 − x3 − x4 = −6 ⎧ x1 − x2 + x3 − x4 = −8 ⎪ x + x2 − x3 + x4 = 19 ⎛ ⎞ c ⎪⎨ ⎜ − ; ; − ;3 ⎟ ⎪ x1 − x2 + x3 + x4 = −1 ⎝ 2 ⎠ ⎪⎩3x1 + x2 − x3 − x4 = −2 ⎧ x1 − x3 + x4 = ⎪ x + x2 − x3 − x4 = d ⎪⎨ ( 0;1; −1;2 ) ⎪5 x1 − 3x4 = −6 ⎪⎩ x1 + x2 + x3 + x4 = ⎧2 x1 + x2 + x4 = ⎪ x − x + 3x = e ⎪⎨ ( −19;26;11; −5) x + x = ⎪ ⎪⎩ x1 + x4 = −24 10 Giải hệ phương trình sau ⎧6 x1 + x2 + x3 = ⎩3 x1 − x2 + x3 = a ⎨ 18 − 15 x1 ⎞ ⎛ ⎜ x1 ; − ; ⎟ 10 ⎠ ⎝ ⎧ x1 + x2 + x3 + x4 = b ⎪⎨2 x1 − x2 + x3 − x4 = ⎪5 x + 3x + x + x = ⎩ ⎧3x1 + x2 + x3 + x4 = c ⎪⎨2 x1 + 3x2 + x3 + x4 = ⎪5 x + x − x + x = ⎩ ⎧ x1 + x2 + x3 = 14 ⎪ ⎪3x1 + x2 + x3 = 10 d ⎪⎨ x1 + x2 + x3 = ⎪2 x + 3x − x = ⎪ ⎪⎩ x1 + x2 = ⎧ x1 + x2 − x3 + x4 + x5 = e ⎪⎨ x1 + 3x2 − x3 + 3x4 + x5 = ⎪ x + 3x − 3x − x = 2 ⎩ x + x − x = − ⎧ ⎪ x − x2 + x3 = f ⎪⎨ ⎪4 x1 + x2 − x3 = −7 ⎪⎩5 x1 − x2 + x3 = ⎧ x1 + x2 + x3 + x4 = ⎪ x + x − x3 − x4 = g ⎪⎨ ⎪ x1 + x2 − x3 + 3x4 = ⎪⎩ x1 + x2 + x3 + x4 = ⎧ x1 + x2 + x3 + x4 = ⎪ x + x3 + 3x4 = h ⎪⎨ ⎪ x1 + x3 + x4 = ⎪⎩ x1 + x2 + x3 + x4 = ⎧ x1 + x2 + 3x3 + x4 = 30 ⎪ − x + x2 − 3x3 + x4 = 10 i ⎪⎨ ⎪ x2 − x3 + x4 = ⎪⎩ x1 + x2 + x3 + x4 = 10 ⎧ x1 − x2 + x3 − x4 = ⎪ x + x + x3 + 3x4 = j ⎪⎨ ⎪2 x1 + x2 + x3 + 10 x4 = 20 ⎪⎩2 x1 − x2 + x3 − x4 = 11 Giải hệ sau ⎧ x1 + x2 + x3 = ⎪ a ⎨2 x1 + x2 + x3 = ⎪3 x + x + x = ⎩ ⎧ x1 − x2 + x3 − x4 = ⎪ b ⎨ x1 + x2 − x3 + x4 = ⎪4 x − x + x + x = ⎩ ⎧ x1 + x2 + x3 = ⎪ c ⎨3 x1 − x2 − x3 = ⎪2 x + 3x + x = ⎩ ⎧3 x1 − x2 + x3 + x4 = ⎪ d ⎨ x1 + x2 − x3 − x4 = ⎪5 x + x − x = ⎩ ⎧2 x1 + x2 + x3 = ⎪3 x + x − x = ⎪ e ⎨ ⎪ x1 + x2 − x3 = ⎪⎩5 x1 + x2 − x3 = ⎧ x1 + x2 + x3 − x5 = ⎪ f ⎨2 x1 + x2 + x3 + x4 + x5 = ⎪ x + 3x + x − x − x = ⎩ ⎧ x1 + x2 + x3 − x4 = ⎪3 x + x + x − x = ⎪ g ⎨ ⎪4 x1 + x2 − x3 + x4 = ⎪⎩3 x1 + x2 + 24 x3 − 19 x4 = ⎧ x1 + x2 − x3 + x4 = ⎪2 x + x + x − x = ⎪ h ⎨ ⎪ x1 + x2 + x3 − x4 = ⎪⎩4 x1 + x2 − x3 + x4 = 12 Xét tính độc lập tuyến tính, phụ thuộc tuyến tính hệ vectơ sau a a1 = (1; 2;1) , a2 = ( 0;1;2 ) , a3 = ( 0;0; ) b a1 = (1;1;0 ) , a2 = (1;0;1) , a3 = (1; −2;0 ) c a1 = (1;3;3) , a2 = (1;1;1) , a3 = ( −2; −4; −4 ) d a1 = (1; −3;0 ) , a2 = ( 3; −3;1) , a3 = ( 2;0;1) e a1 = ( 2;3;1) , a2 = (1;1;1) , a3 = (1; 2;0 ) f a1 = (1; 2;3; ) , a2 = (1;3; 2; ) , a3 = ( 2;5;5;8 ) g a1 = (1;1;1;1) , a2 = ( 0;1;1;1) , a3 = ( 0;0;1;1) , a4 = ( 0;0;0;1) h a1 = ( 2;0;3) , a2 = ( 5; −1;7 ) , a3 = ( −1; 2; −1) i a1 = ( 0; −2;3) , a2 = ( 3; 2; −1) , a3 = ( 3;0;2 ) j a1 = ( −1; 2;3) , a2 = ( 2;0; −1) , a3 = ( −5;6;11) 13 Hãy biểu diễn vectơ a thành tổ hợp tuyến tính vectơ lại a a = (1; 2;0 ) , b = (1;2; −3) , c = ( 2;5; −1) , d = ( 0;1; ) b a = ( 0;0;0 ) , b = ( 2;3;3) , c = ( 4;9;1) , d = (1;3; −1) c a = (1;1;1) , b = (1;2; ) , c = ( 2;1; ) , d = ( 3; 4; ) 14 Trong không gian vectơ \3 , xét xem W có khơng gian vectơ \3 không? a W = { x = ( x1; x2 ; x3 ) / x1 + x2 + x3 = 0} b W = { x = ( x1; x2 ; x3 ) / x1 − x2 + x3 = 1} c W = { x = ( x1 ; x2 ; x3 ) / x1 + x2 = x3} d W = { x = ( x1; x2 ; x3 ) / x1 + x2 + = x3} 15 Tìm vectơ sinh khơng gian nghiệm hệ phương trình ⎧ x1 + x2 + x3 + x4 + x5 = a ⎨ ⎩2 x1 + x2 + x3 + x4 + x5 = ⎧ x1 + x2 + x3 + x4 + x5 = b ⎨ ⎩ x1 + x2 + x3 + x4 + x5 = ⎧ x1 + x2 + x3 + x4 + x5 = ⎪ c ⎨2 x1 + x2 + x3 + x4 + x5 = ⎪3 x + x + x + x + x = ⎩ 10 16 Trong không gian vectơ \ , hệ vectơ sau có sở khơng ? Nếu có, tìm tọa độ vectơ a theo sở a a1 = (1;1;1) , a2 = (1; 2;3) , a3 = ( 2; 2;4 ) , a = ( 0;0; ) b a1 = ( 2;1; ) , a2 = ( 3;2;3) , a3 = (1;1; ) , a = (1;2;3) c a1 = ( −1;1; ) , a2 = ( 0; 2;1) , a3 = (1;1;1) , a = ( 2; 2; ) 17 Tìm tất trị riêng vá vectơ riêng tương ứng ma trận sau ⎡ −1 ⎤ a A = ⎢⎢ ⎥⎥ ⎢⎣1 −3 3⎥⎦ ⎡2 0⎤ b A = ⎢⎢ ⎥⎥ ⎢⎣1 −2 3⎥⎦ ⎡ −2 −1⎤ c A = ⎢⎢1 −1⎥⎥ ⎢⎣ −2 ⎥⎦ ⎡6 d A = ⎢⎢ ⎢⎣ ⎡0 e A = ⎢⎢ ⎢⎣ −2 −2 −3⎤ −2 ⎥⎥ −2 −1⎥⎦ 1⎤ ⎥⎥ 3⎥⎦ ⎡ −1 ⎤ f A = ⎢⎢ −2 2 ⎥⎥ ⎢⎣ −2 ⎥⎦ ⎡ −3 ⎤ g A = ⎢⎢ −4 ⎥⎥ ⎢⎣ −4 ⎥⎦ ⎡ −1 h A = ⎢⎢ ⎢⎣ −4 ⎡7 −2 i A = ⎢⎢8 −1 ⎢⎣ −2 2⎤ ⎥⎥ ⎥⎦ −6 ⎤ −8⎥⎥ −3⎥⎦ ⎡ 0⎤ j A = ⎢⎢ −1 ⎥⎥ ⎢⎣ −1 1 ⎥⎦ 11 ⎡1 k A = ⎢⎢ −1 ⎢⎣ −1 ⎡2 l A = ⎢⎢1 ⎢⎣1 −2 0⎤ ⎥⎥ ⎥⎦ 0⎤ ⎥⎥ 3⎥⎦ ⎡ −1⎤ m A = ⎢⎢ −1⎥⎥ ⎢⎣ 1 ⎥⎦ ⎡ −1 ⎤ n A = ⎢⎢ 1 ⎥⎥ ⎢⎣ −1 ⎥⎦ 12 ... x + x + x − x = ⎪ h ⎨ ⎪ x1 + x2 + x3 − x4 = ⎪⎩4 x1 + x2 − x3 + x4 = 12 Xét tính độc lập tuyến tính, phụ thuộc tuyến tính hệ vectơ sau a a1 = (1; 2;1) , a2 = ( 0;1;2 ) , a3 = ( 0;0; ) b a1 = (1;1;0... ) j a1 = ( −1; 2;3) , a2 = ( 2;0; −1) , a3 = ( −5;6;11) 13 Hãy biểu diễn vectơ a thành tổ hợp tuyến tính vectơ lại a a = (1; 2;0 ) , b = (1;2; −3) , c = ( 2;5; −1) , d = ( 0;1; ) b a = ( 0;0;0... ⎡1 ⎤ ⎢6 ⎥ ⎛ ⎡1 4⎤ ⎥ ⎜ AB = ⎡ 28 27 ⎤ ⎟⎞ , B=⎢ c A = ⎢ ⎢15 14 13⎥ ⎥ ⎢1 −1⎥ ⎝ ⎣ ⎦⎠ ⎣ −2 ⎦ ⎢ ⎥ ⎣3 ⎦ Tính định thức sau 0 a 0 b a Δ = ( abcd ) c 0 d 0 b Δ = c Δ = 1 4 3 2 1 1 3 −1 d Δ = e Δ = 1 ( 0)